The Poisson equation from non-local to local

U. Biccari, V. Hernández-Santamaría The Poisson equation from non-local to local, Electronic Journal of Differential Equations, Vol. 2018 (2018), No. 145, pp. 1-13. DOI: arXiv:1801.09470

Abstract: We analyze the limit behavior as $s\to 1^-$ of the solution to the fractional Poisson equation $(-\Delta)^s u_s=f_s , x\in\Omega$ with homogeneous Dirichlet boundary conditions $u_s\equiv 0 , x\in\Omega^c$. We show that $\lim_{s\to 1^-} u_s =u$ with $-\Delta u =f , x\in\Omega$ and $u=0 , x\in\partial\Omega$. Our results are complemented by a discussion on the rate of convergence and on extensions to the parabolic setting.

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